Slove: 2logbase5(x+2) = 2 + log base5(x-2)

assuming base 5 throughout,

2log(x+2) = 2 + log(x-2)
log[(x+2)^2 / (x-2)] = 2
(x+2)^2 = 25(x-2)
x = 3 or 18

To solve the equation 2logbase5(x+2) = 2 + log base5(x-2), we can start by applying several logarithmic properties.

First, we can use the product rule of logarithms, which states that log base a (mn) = log base a (m) + log base a (n).

Using the product rule, we can rewrite the equation as: log base 5 ((x+2)^2) = 2 + log base 5 (x-2).

Next, we can apply the power rule of logarithms, which states that log base a (m^k) = k * log base a (m).

Using the power rule, we can rewrite the equation as: log base 5 (x+2)^2 = log base 5 (x-2) + 2.

Now, we can use the property of logarithms that if two logarithms with the same base are equal, then their arguments must be equal as well.

Applying this property, we have: (x+2)^2 = (x-2) * 5^2.

Simplifying further, we get: (x+2)^2 = 25(x-2).

Expanding the left side of the equation, we have: x^2 + 4x + 4 = 25x - 50.

Rearranging the terms, we get: x^2 - 21x + 54 = 0.

Now, we can solve this quadratic equation by factoring or using the quadratic formula.

Factoring the equation, we have: (x-3)(x-18) = 0.

Setting each factor equal to zero, we get: x-3 = 0 or x-18 = 0.

Solving each equation, we have: x = 3 or x = 18.

Therefore, the solutions to the equation 2logbase5(x+2) = 2 + log base5(x-2) are x = 3 and x = 18.